Properties of geometrical realizations of substitutions associated to a family of Pisot numbers

Abstract

In this thesis we study some properties of the geometrical realizations of the dynamical systems that arise from the family of Pisot substitutions: 1 → 12 IIn :2 → 13 : (n −1) → 1n n → 1 for n a positive integer greater than 2. In chapter 1 we compute the Holder exponent of the Arnoux map, which is the semiconjugacy between the geometrical realization of (Ω ό), the dynamical system of this substitution, in the circle (SI, f) and the n – 1 dimensional torus (Tn-I, T). Also in this chapter we introduce the notion of the standard partition in the symbolic space Ω and in its geometrical realizations. The cylinders of this partition are classified according to their structure. In chapter 2 we construct a geodesic lamination on the hyperbolic disk associated to this standard partition and a transverse measure on the lamination. The interval exchange map f and the contraction h induce maps F and H on the lamination, respectively. The map .F preserves the transverse measure and H contracts it. In chapter 3 we compute the Hausdorff dimension of the boundary of w,. the fundamental domain of the torus T2 obtained by the realization of the symbolic Ω space that arises from the substitution II3. As a corollary we compute the Hausdorff dimension of the pre-image·of the boundary of w under the Arnoux map. We also describe the identifications on the boundary of w that make it a fundamental domain of the two dimensional torus. In chapter 4 we study some relationships between the dynamical systems of this family of substitutions. We describe how the dynamics of the systems of this family corresponding to lower dimensions - i.e. the parameter n in the definition of IIn - present in systems of higher dimensions. Also we study the realization of this property in the interval